Completely hyperexpansive operators with finite rank defect operator and de Branges-Rovnyak spaces

TL;DR

This paper characterizes cyclic, analytic, completely hyperexpansive operators with finite rank defect operators using de Branges-Rovnyak space symbols, extending understanding of their structure and relation to Dirichlet-type spaces.

Contribution

It provides a new characterization of such operators in terms of de Branges-Rovnyak space symbols, building on Aleman's model.

Findings

Characterization of cyclic, analytic, completely hyperexpansive operators with finite rank defect.

Connection established between these operators and de Branges-Rovnyak space symbols.

Extension of previous models to a broader class of operators.

Abstract

The process of identifying a Dirichlet-type space $D(\mu)$ for a positive, Borel measure $\mu$, supported on the unit circle $\mathbb T,$ with a de Branges-Rovnyak space was initiated by Sarason. A characterization of the symbol for a de Branges-Rovnyak spaces for which the shift operator is a $2$-isometry, was provided in an article by Kellay and Zarrabi. In this paper, capitalizing on the Aleman's model for the cyclic, analytic, completely hyperexpansive operators, we provide a characterization of cyclic, analytic, completely hyperexpansive operator with finite rank defect operator in terms of the symbol for a de Branges-Rovnyak space.